Overview
A function maps each input to exactly one output. Know the notation, inverses and composite functions.
Key terms
- Notation: f(x) means "the function f applied to x".
- Domain = set of allowed inputs; range = set of possible outputs.
- A one-one function gives a different output for every input (needed for an
inverse).
Composite functions
fg(x) means "do g first, then f": fg(x) = f(g(x)). Order matters — fg ≠ gf in general.
Inverse functions
The inverse f⁻¹(x) reverses f. To find it: write y = f(x), swap x and y, and make y the subject. The graph of f⁻¹ is the reflection of f in the line y = x.
The modulus function
|x| gives the size of x without its sign, so |x| ≥ 0 always.
Worked example
If f(x) = 2x + 1, find f⁻¹(x): y = 2x + 1 → x = (y − 1) ÷ 2, so f⁻¹(x) = (x − 1)/2.
Common mistakes
- Doing fg as "f then g" — it's actually g then f.
Exam tips
- Only a one-one function has an inverse — restrict the domain if needed.